Question Video: Determining Triangle Types from Side Lengths | Nagwa Question Video: Determining Triangle Types from Side Lengths | Nagwa

Question Video: Determining Triangle Types from Side Lengths Mathematics • Second Year of Preparatory School

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Given that 𝐴𝐶 = 12 cm, 𝐵𝐶 = 6 cm, and 𝐴𝐵 = 13 cm, what type is the largest angle?

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Video Transcript

Given that 𝐴𝐶 is equal to 12 centimeters, 𝐵𝐶 is equal to six centimeters, and 𝐴𝐵 is equal to 13 centimeters, what type is the largest angle?

In this question, we are given the lengths of three sides in a triangle and asked to determine the type of the largest internal angle in the triangle. To do this, we will compare the side lengths to those in a right triangle.

We note that side 𝐴𝐵 is the longest with length 13 centimeters. In a right triangle, the Pythagorean theorem tells us that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. So 𝐴𝐵 squared is equal to 𝐴𝐶 squared plus 𝐵𝐶 squared.

We can also recall that the Pythagorean inequality theorem allows us to determine the type of the largest angle by comparing the square of the length of the longest side to the sum of the squares of the lengths of the two shorter sides. In this case, if 𝐴𝐵 squared is greater than 𝐴𝐶 squared plus 𝐵𝐶 squared, then the angle at 𝐶 is obtuse. And if 𝐴𝐵 squared is less than 𝐴𝐶 squared plus 𝐵𝐶 squared, then the angle at 𝐶 is acute.

We can calculate both of these values for the given triangle. We have that 𝐴𝐵 squared is 13 squared, which is equal to 169. And 𝐴𝐶 squared plus 𝐵𝐶 squared is equal to 12 squared plus six squared, which is equal to 180. We can see that the square of the length of the longest side is less than the sum of the squares of the two shorter sides. So the largest angle in the triangle must be an acute angle.

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